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Space form

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In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

Reduction to generalized crystallography

The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M n {\displaystyle M^{n}} with curvature K = − 1 {\displaystyle K=-1} is isometric to H n {\displaystyle H^{n}} , hyperbolic space, with curvature K = 0 {\displaystyle K=0} is isometric to R n {\displaystyle R^{n}} , Euclidean n-space, and with curvature K = + 1 {\displaystyle K=+1} is isometric to S n {\displaystyle S^{n}} , the n-dimensional sphere of points distance 1 from the origin in R n + 1 {\displaystyle R^{n+1}} .

By rescaling the Riemannian metric on H n {\displaystyle H^{n}} , we may create a space M K {\displaystyle M_{K}} of constant curvature K {\displaystyle K} for any K < 0 {\displaystyle K<0} . Similarly, by rescaling the Riemannian metric on S n {\displaystyle S^{n}} , we may create a space M K {\displaystyle M_{K}} of constant curvature K {\displaystyle K} for any K > 0 {\displaystyle K>0} . Thus the universal cover of a space form M {\displaystyle M} with constant curvature K {\displaystyle K} is isometric to M K {\displaystyle M_{K}} .

This reduces the problem of studying space forms to studying discrete groups of isometries Γ {\displaystyle \Gamma } of M K {\displaystyle M_{K}} which act properly discontinuously. Note that the fundamental group of M {\displaystyle M} , π 1 ( M ) {\displaystyle \pi _{1}(M)} , will be isomorphic to Γ {\displaystyle \Gamma } . Groups acting in this manner on R n {\displaystyle R^{n}} are called crystallographic groups. Groups acting in this manner on H 2 {\displaystyle H^{2}} and H 3 {\displaystyle H^{3}} are called Fuchsian groups and Kleinian groups, respectively.

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